Due to the importance of studying the change in glucose which affected by obesity and results in a negative effect on the health of the body, In this paper, glucose metabolism model was developed to study the effect of obesity. The bifurcations and related dynamical behaviors of our model were studied completely . For the non-delay model, it is shown that the model undergoes co-dimension 1 bifurcation such as transcritical, Hopf and saddle bifurcations, and co-dimension 2 bifurcation such as generated hopf, zero-hopf and cusp bifurcations. On the other hand, we consider there is a lag for responding of Insulin sensitivity under obesity effect. Therefore, the time delay $\tau$ is deemed as an additional factor in the glucose metabolism model. We set the condition for the existence of the delay parameter value for which the model displays a Hopf bifurcation. Finally, numerical simulations are given to support theoretical analysis and also to show the complex dynamical behaviors including families of periodic curves. The described dynamics of this model can obviously illustrate several clinical observations for obesity effect on glucose level, oscillation and disorder, supporting its potential role in control of blood glucose.

由于研究肥胖影响的葡萄糖变化对身体健康产生负面影响的重要性，本文建立了葡萄糖代谢模型来研究肥胖的影响。全面研究了模型的分叉及其相关的动力学行为。对于非延迟模型，表明模型经历了共维1分叉，例如跨临界，Hopf和鞍形分叉，以及共维2分叉，例如生成的Hopf，零跳和尖顶分叉。另一方面，我们认为肥胖效应下对胰岛素敏感性的反应存在滞后性。因此，时间延迟$\tau$被认为是葡萄糖代谢模型中的另一个因素。我们设置了存在延迟参数值的条件，为此模型显示了Hopf分支。最后，给出了数值模拟，以支持理论分析并显示出复杂的动力学行为，包括周期曲线族。所描述的模型动力学可以明显说明肥胖对血糖水平，振荡和失调的临床观察，支持其在控制血糖方面的潜在作用。